Yesterday my students had a homework assignment from their textbook where they had to graph a couple conic sections that had been rotated. One of them was particularly tedious (3x^2 - 2sqrt(3)xy +y^2 +2x +2sqrt(3)y = 0, so there are usually lots of questions about it. I will have a student who got the correct answer present their work on the document camera.
One thing I tell students is that the xy-term should always go away when we convert it to a new equation in the x'y'-plane. That is the whole point of this process - to make the equation into a form we are familiar with so that we can graph like the conics that are horizontal or vertical.
Here is the unit review that I give my students and have them work on for the rest of the hour. They are in their table groups and so they can talk through their thoughts while working.
In this video I discuss some of the problems on the worksheet.